Instructor: Alexander Stoytchev CprE 185: Intro to Problem Solving (using C)

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Chapter 10 (Recursion)

Recursion (part 2) CprE 185: Intro to Problem Solving Iowa State University, Ames, IA Copyright © Alexander Stoytchev

Examples of Recursion

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The von Koch Curve and Snowflake [

Divide it into three equal parts [

Replace the inner third of it with an equilateral triangle [

Repeat the first two steps on all lines of the new figure [

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Quick Review of the Last Lecture (with more in depth examples)

© 2004 Pearson Addison-Wesley. All rights reserved Recursion Recursion is a fundamental programming technique that can provide an elegant solution certain kinds of problems.

© 2004 Pearson Addison-Wesley. All rights reserved Recursive Thinking A recursive definition is one which uses the word or concept being defined in the definition itself When defining an English word, a recursive definition is often not helpful But in other situations, a recursive definition can be an appropriate way to express a concept Before applying recursion to programming, it is best to practice thinking recursively

© 2004 Pearson Addison-Wesley. All rights reserved Circular Definitions Debugger – a tool that is used for debugging

© 2004 Pearson Addison-Wesley. All rights reserved Recursive Definitions Consider the following list of numbers: 24, 88, 40, 37 Such a list can be defined as follows: A LIST is a: number or a: number comma LIST That is, a LIST is defined to be a single number, or a number followed by a comma followed by a LIST The concept of a LIST is used to define itself

© 2004 Pearson Addison-Wesley. All rights reserved Recursive Definitions The recursive part of the LIST definition is used several times, terminating with the non-recursive part: number comma LIST 24, 88, 40, 37 number comma LIST 88, 40, 37 number comma LIST 40, 37 number 37

© 2004 Pearson Addison-Wesley. All rights reserved Infinite Recursion All recursive definitions have to have a non- recursive part If they didn't, there would be no way to terminate the recursive path Such a definition would cause infinite recursion This problem is similar to an infinite loop, but the non-terminating "loop" is part of the definition itself The non-recursive part is often called the base case

© 2004 Pearson Addison-Wesley. All rights reserved Recursive Definitions N!, for any positive integer N, is defined to be the product of all integers between 1 and N inclusive This definition can be expressed recursively as: 1! = 1 N! = N * (N-1)! A factorial is defined in terms of another factorial Eventually, the base case of 1! is reached

© 2004 Pearson Addison-Wesley. All rights reserved Recursive Definitions 5! 5 * 4! 4 * 3! 3 * 2! 2 * 1!

Example: Factorial_Iterative.c int factorial(int a) { int i,result; result=1; for(i=1;i<=a;i++) result=result*i; return result; }

Example: Factorial_Recursive.c int factorial(int a) { if(a==1) return 1; else return a*factorial(a-1); }

Recursive Execution 6! (6 * 5!) (6 * (5 * 4!)) (6 * (5 * (4 * 3!))) (6 * (5 * (4 * (3 * 2!)))) (6 * (5 * (4 * (3 * (2 * 1!))))) (6 * (5 * (4 * (3 * (2 * (1 * 0!)))))) (6 * (5 * (4 * (3 * (2 * (1 * 1)))))) (6 * (5 * (4 * (3 * (2 * 1))))) (6 * (5 * (4 * (3 * 2)))) (6 * (5 * (4 * 6))) (6 * (5 * 24)) (6 * 120) 720

Recursive Programming A function in C can invoke itself; if set up that way, it is called a recursive function The code of a recursive function must be structured to handle both the base case and the recursive case Each call to the function sets up a new execution environment, with new parameters and local variables As with any function call, when the function completes, control returns to the function that invoked it (which may be an earlier invocation of itself)

© 2004 Pearson Addison-Wesley. All rights reserved Recursive Programming Consider the problem of computing the sum of all the numbers between 1 and any positive integer N This problem can be recursively defined as:

Example: Sum_Iterative.c int sum(int a) { int i,result; result=0; for(i=1;i<=a;i++) result=result+i; return result; }

Example: Sum_Recursive.c int sum(int a) { if(a==1) return 1; else return a+sum(a-1); }

© 2004 Pearson Addison-Wesley. All rights reserved Recursive Programming // This function returns the sum of 1 to num int sum (int num) { int result; if (num == 1) result = 1; else result = num + sum (n-1); return result; }

sum(1); sum(1)sum(2) Recursive Control Flow In Recursive calls methods can call themselves, but typically with different arguments each time return 1;

Recursive Control Flow In Recursive calls methods can call themselves, but typically with different arguments each time sum(2) sum(1) sum(1); sum(2); sum(3) main() return 1; sum(3);

© 2004 Pearson Addison-Wesley. All rights reserved Recursive Programming main sum sum(3) sum(1) sum(2) result = 1 result = 3 result = 6

Fibonacci Sequence #include int fib(int n) { int ret; if (n < 2) { ret = 1; } else{ ret= fib(n-1) + fib(n-2); } return ret; } int main() { int i; for(i=0;i<10;i++) printf("%d ", fib(i)); system("pause"); }

Recursion: Fibonacci Numbers The sequence: {0,1,1,2,3,5,8,13,...}

Mathematical notation v.s. C code int fib(int n) { if(n <= 1) return n; //base case else return fib(n-1) + fib(n-2); }

Execution Trace

Mystery Recursion void mystery1(int a, int b) { if (a <= b) { int m = (a + b) / 2; printf(“%d “, m); mystery1(a, m-1); mystery1(m+1, b); } } int main() { mystery1(0, 5); system(“pause”); }

Think of recursion as a tree …

… an upside down tree

(a=0, b=5) (a=0, b=1)(a=3, b=5) (a=0, b=-1)(a=1, b=1)(a=3, b=3)(a=5, b=5) (a=1, b=0)(a=2, b=1)(a=3, b=2)(a=4, b=3)(a=6, b=5)(a=5, b=4)

(a=0, b=5) (a=0, b=1)(a=3, b=5) (a=0, b=-1)(a=1, b=1)(a=3, b=3)(a=5, b=5) (a=1, b=0)(a=2, b=1)(a=3, b=2)(a=4, b=3)(a=6, b=5)(a=5, b=4) 2 m=(a+b)/2 => print 2

(a=0, b=5) (a=0, b=1)(a=3, b=5) (a=0, b=-1)(a=1, b=1)(a=3, b=3)(a=5, b=5) (a=1, b=0)(a=2, b=1)(a=3, b=2)(a=4, b=3)(a=6, b=5)(a=5, b=4)

(a=0, b=5) (a=0, b=1)(a=3, b=5) (a=0, b=-1)(a=1, b=1)(a=3, b=3)(a=5, b=5) (a=1, b=0)(a=2, b=1)(a=3, b=2)(a=4, b=3)(a=6, b=5)(a=5, b=4) Traversal Order

Questions?

Other stuff for the labs this week

Stack Animation animations/tsang/html/STACK/stack1024.html

Memory Organization [

The stack during a recursive call to factorial

The stack during a recursive call to gcd [

Recursive Programming Note that just because we can use recursion to solve a problem, doesn't mean we should For instance, we usually would not use recursion to solve the sum of 1 to N problem, because the iterative version is easier to understand However, for some problems, recursion provides an elegant solution, often cleaner than an iterative version You must carefully decide whether recursion is the correct technique for any problem

Maze Traversal We can use recursion to find a path through a maze From each location, we can search in each direction Recursion keeps track of the path through the maze The base case is an invalid move or reaching the final destination

Traversing a maze

Towers of Hanoi The Towers of Hanoi is a puzzle made up of three vertical pegs and several disks that slide on the pegs The disks are of varying size, initially placed on one peg with the largest disk on the bottom with increasingly smaller ones on top The goal is to move all of the disks from one peg to another under the following rules:  We can move only one disk at a time  We cannot move a larger disk on top of a smaller one

Towers of Hanoi Original ConfigurationMove 1Move 3Move 2

Towers of Hanoi Move 4Move 5Move 6Move 7 (done)

Animation of the Towers of Hanoi WangApr01/RootWang.html

Towers of Hanoi An iterative solution to the Towers of Hanoi is quite complex A recursive solution is much shorter and more elegant

Indirect Recursion A method invoking itself is considered to be direct recursion A method could invoke another method, which invokes another, etc., until eventually the original method is invoked again For example, method m1 could invoke m2, which invokes m3, which in turn invokes m1 again This is called indirect recursion, and requires all the same care as direct recursion It is often more difficult to trace and debug

Indirect Recursion m1m2m3 m1m2m3 m1m2m3

THE END